Chapter 15: Homework Problems

Transcription

1 Chapter 15: Homework Problems Going back to the Iron Man problem you encountered in Chapter 6 (problem 6.1), use the calculated value of T max to determine the acceleration that will be generated if Iron Man initiates maximum thrust in each boot (heading straight up). If he starts at 0 m/s and accelerates at this maximum rate, how high in the air will he be when he reaches a speed of 300 ft./s. Iron Man c and TM, Marvel Entertainment, LLC all rights reserved and used with permission Top fuel dragsters are some of the fastest accelerating vehicles on earth (and compare favorably with the space shuttle). After finishing their quarter-mile run, a parachute deploys and they come to a stop within a few seconds. As a result of the extreme deceleration, drivers have been known to detach their retinas. Just thought that might be interesting. For the purposes of this problem, assume that the dragster crosses the finish line at a top speed of 333 mph and it has a weight of 2,225 lbs. a) If the acceleration is constant over the entire run, what is its magnitude? How long does it take the driver to complete the course? b) Assume that, after the dragster crosses the finish line, the engine cuts off and a parachute opens generating a drag force, F D = Av x. Draw a free body diagram of the dragster. c) If it takes 17.5 seconds for the dragster to reach 1/10 of its top speed, what is the value of the constant, A.

2 Chapter 15: Homework Problems d) What is the peak magnitude of the acceleration experienced by the driver during the race?

3 Chapter 15: Homework Problems The Battleship Missouri played an important role in the Battle of the Pacific during World War II and was the location of the official Japanese surrender. You may assume that the battleship weights 50,000 tons and that the water exerts a drag force of the form F = Av 2, where A is a constant. We also know that, to keep the battleship moving at a constant speed of 40 ft./s, the propellers must generate a total thrust of 250,000 lbs. a) Determine the constant, A, where A is a constant with units of lbs. s 2 /ft 2. b) If the ship is traveling at 40 ft./s and the propellers are suddenly powered down, how far does it travel before it reaches a speed of 10 ft./s.

4 Chapter 15: Homework Problems 14.4 Ermes Zamperla, a human cannonball, performed his stunt up to four times per day. In February of 2002, he miscalculated and, instead of landing on a cushion 75 ft. from the cannon, he actually traveled 100 ft., breaking both legs and bones in his back. If he takes off at an angle of 50 o, what was his initial velocity (you may neglect wind resistance). What should his initial velocity have been? What is the percent difference between the two? Would you consider launching yourself in this way?

5 Chapter 15: Homework Problems The Mythbusters are led by Adam Savage and Jamie Hyneman (also including Tory Belleci, Kari Byron, Grant Imahara, Jessi Combs, and Buster). For some reason, Adam and Jamie decided to see if it was possible for an explosion caused by methane detonation 16 in the sewer system to launch a sewer caps (a.k.a. manhole cover). They first used a scaled down version to see if it was even possible. Once they decided that it was, they built a full size sewer system 17, filled it with methane, and set it off (please do not try this on campus or at home!). The sewer caps traveled 150 ft. straight up into the air. If each one weighs 150 lbs., determine the initial velocity required to launch the sewer caps. If the sewer caps were launched at the same initial velocity, but at a 45 o angle, how far would they travel? 16 It turns out that their experiment was not able to cause a detonation, just a massive conflagration. To the untrained eye, there isn t much difference between a detonation and a conflagration, but, had it detonated, the sewer caps would have gone much further. 17 see

6 Chapter 15: Homework Problems 14.6 One way to measure friction coefficients is to use a simple beam and placing a disk of mass, m on it. You then raise one end slowly until the disk starts to slide down. The angle at which the disk starts sliding allows you to determine the coefficient of static friction, µ s. The time it takes for the disk to slide down, allows you to determine the coefficient of kinetic friction µ k. L θ If the disk has a mass of 0.15 kg, the length of the beam is 0.4 m, the disk starts to slide when θ is 55 o and it takes 10 seconds for the disk to slide down, find the coefficients of static and kinetic friction.

7 Chapter 15: Homework Problems In 1967 the Henderson Brothers (Bill and Jim) bought a 1933 Plymouth to drive around rural Iowa. For a car that was already 34 years old, the top speed was pretty good although for some time it only turned left. Other challenges included the lack of floor boards, brakes, a holder for the battery 18, or anything to prevent flames from shooting out the side of the engine while it was running. The 33 Plymouth had suicide doors (as shown below) and, to stop the car, they would have to throw the doors open, and each person would put a foot out to drag on the ground and provide friction in the hope that it would be sufficient to stop the car before they ran into their mother s giant oak tree at the end of the drive way 19. a) If the Henderson brothers approach the drive way at 15 m.p.h. and both of them put a foot out to brake the car, do they have enough room to stop? If so, you are done with the problem. b) If not, determine how fast they are going when they hit the oak tree. You may assume that the coefficient of kinetic friction between their feet and the ground is 0.8 and the maximum normal force they can generate is approximately one half their body weight (you may assume body weight is 180 lbs.). You may further assume that the car weighs 1,200 lbs. The driveway was approximately 30 yards long. 18 The younger Henderson brother was often tasked with holding the battery in his lap. 19 It should go without saying that you should not try this - at home or anywhere else.

8 Chapter 15: Homework Problems 14.8 A 1.2 kg ball moves along a curve whose path is given by y = (12 x 2 ) where x and y are measured in feet. At x = 1.0ft., the ball is moving in the x-direction at a constant rate of +2 ft./s. Draw the basis vectors for path coordinates on the figure below. a) What are the velocity and acceleration vectors when x = 1.0ft.? b) What force is required at this instant to cause this motion? Please write your answer as a vector in Cartesian coordinates. y 12 x

9 Chapter 15: Homework Problems A 150 gram bead is constrained to move along a path given by y = 0.5x 3 where x and y are measured in meters. At x = 5m, the bead is moving in the x direction at a constant rate of 0.5 m/s. Draw the basis vectors for path coordinates on the figure below. What are the velocity and acceleration vectors when x = 5m? Determine the forces acting on the bead at this instant. Please write your answer as a vector in Cartesian coordinates. The gravitational force points downward.

10 Chapter 15: Homework Problems A car traveling around a curve at a speed of 100 km/h is decelerating at a constant rate of 10m/s 2. If the radius of curvature is 500 m, calculate the magnitude of the total acceleration at this instant. Then determine the normal (friction) and tangential forces that result. If the coefficient of static friction between the road and the tires is 0.75, will the car stay on the road? vn vt ρ

11 Chapter 15: Homework Problems Banked curves. The Indianapolis Motor Speedway is a 2.5 mile track with four 1/4 mile turns and two long straightaways that are 5/8 mile each and two shorter straightaways (1/8 mile each). Sketch the tangential and normal coordinates for an Indy car moving through one of the turns at constant speed. If the banking angle is approximately 9 o, what constant speed would be required for the car to slide across the track? The coefficient of static friction between the car and the track is 0.7. The car weighs 1,200 lbs. θ

12 Chapter 15: Homework Problems This time our rocket is fired straight up initially, but follows a curved path afterwards. At one particular instant, the RADAR system notes that the radius, r, is 90,000 ft., ṙ = 2000ft./s, θ = 0.01rad/s, and θ = rad/s 2. Sketch the flight path for the rocket. Calculate the radius of curvature at this instant. You may assume that θ = 75 o and r = 20ft./s 2. Then determine the rate of change of speed. If the rocket engines generate thrust force in the tangential direction, what is the thrust in the engines? The mass of the rocket is 2.75x10 6 kg. USA y r θ x

13 Chapter 15: Homework Problems There is a 5 kg package that is being pushed up a smooth hill (you may assume that this is a processing step similar to the kind used by FedEx to move packages from one place to another. Check out the video: If the force, F pushes in the tangential direction (always) and the profile of the hill is given by y = x2 where x and y are measured in meters, determine the magnitude of F 4 and the normal force. You may assume that the speed of the package is constant at 5 m/s and a = 3m. y a v y=f(x) F x

14 Chapter 15: Homework Problems There is a 5 kg package that is being pushed up a smooth hill (you may assume that this is a processing step similar to the kind used by FedEx to move packages from one place to another. Check out the video: If the force, F pushes in the tangential direction (always) and the profile of the hill is given by y = x 3 where x and y are measured in meters, determine the magnitude of F and the normal force. You may assume that the speed of the package is constant at 5 m/s and a = 3m. y a v y=f(x) F x

15 Chapter 15: Homework Problems There is a 2 lbs. collar pulled to the right along the curved rod ABC whose shape in the horizontal plane is given by y = (3x 2 +2) where x and y are measured in ft. Arm D, which produces this motion has a constant speed of 20 ft./s and a = 3ft. There is also an external force applied to the collar that is a constant 5 lbs. always in the tangential direction. Determine the force exerted by the slider (B) and the force exerted by the rod (D). C y A v F B y=f(x) a D x

16 Chapter 15: Homework Problems Spider-Man (height 5 10, 165 lbs.) spends a lot of time swinging around New York City. Below is an image as he just starts his swing. The webline attaches to a building approximately 200 ft. away and makes an initial angle of 35 o with the horizontal. If he starts with zero velocity, how fast will he be moving at the bottom of his swing? What is the maximum force in the webline? Spider-Man c and TM, Marvel Entertainment, LLC all rights reserved and used with permission.

17 Chapter 15: Homework Problems Particle A, having a mass of 4 kg, is constrained to move within a smooth slot in a rotating arm. The arm rotates at a constant rate of ω = 7.5rad/s about point O within a horizontal plane (i.e. gravity acts into the page. A constant force F A = 150N acts on the particle in the outward radial direction. The radial distance from O to A is given by the distance r in the figure below. a) Draw a free body diagram of particle A. Show the unit vectors, e r and e θ in your FBD. b) Write down Newton s second law for the particle. c) If ṙ = 4m/s when r = 0.2m, determine the value of ṙ when r = 0.5m. d) Determine the contact force F S of the slot on particle A when r = 0.5m.

18 Chapter 15: Homework Problems Rotating pendulum. A pendulum is rotating at a constant speed and a constant angle, β = 35 o. If L = 1m, and a mass of 2.3 kg. Determine the tension in the cable. O g h β L r v P m

19 Chapter 15: Homework Problems The Gravitron is an interesting ride. You start off by leaning against the outside wall facing the inside (the ride is a giant cylinder). The operator then starts it spinning, eventually reaching a constant angular velocity 20. If the coefficient of friction between the rider and the wall is 0.68 and the radius of the cylinder is 7 m, determine the minimum constant angular velocity required to keep a person from sliding down when the floor drops out beneath them At the Missouri State Fair in 1991, a Gravitron actually spun itself apart which causing injuries to seven children. They have since been modified to improve their safety. 21 For most versions of the ride, the floor doesn t actually drop out, it just feels like it.

20 Chapter 15: Homework Problems A 2 kg block slides over a smooth smooth rigid arm AB, which is rotating about a fixed pin joint A in the vertical plane at a constant 2 rad/s counterclockwise. A radial force F (shown) is applied in such a manner as to cause the block to move inward (towards point A) at a constant 0.8 m/s. For the instant at which θ = 30 o and r = 0.45m determine the following: (a) Calculate the acceleration of the block in cylindrical coordinates, please write your answer in vector form, (b) Draw the free body diagram using the block provided below, clearly indicating the direction of the cylindrical unit vectors, (c) Write the kinetics equations of motion for the radial (r) and transverse, θ, directions in symbolic form (i.e. do not substitute numbers into the equations until part (d)), and (d) Calculate the applied force F (assuming negligible friction) and the normal force N due to arm AB using the information provided above. g F B A θ

21 Chapter 15: Homework Problems The Ferris Wheel rotates with a constant angular velocity of 5 o /s. Suppose that the rider starts at θ = 0 o (also y = 5m, r = 20m). What force must be exerted on the rider in order to ensure that they move with the Ferris Wheel? If the car goes from 0 to 60 m.p.h. at a constant rate in 7 seconds, what force does the car seat exert on the passenger? ω A θ r y x B v

22 Chapter 15: Homework Problems If W 2 weighs 30 Newtons and W 1 weighs 60 Newtons, and the system is released from rest, what is the initial acceleration of each block? θ W2 W1

23 Chapter 15: Homework Problems Blocks A and B (having masses of m A and m B, respectively) are connected by the cable-pulley system shown. Block A rests on a rough, horizontal surface with µ s being the coefficient of static friction between A and the horizontal surface. A horizontal force P acts to the left on block A, as shown below in the figure. Consider the pulleys to be ideal. (a) Determine the minimum force P required to hold the system in equilibrium. (b) Determine the minimum force P required to initiate motion of block A to the left. (c) Double the force you found in part (b). What is the acceleration? If that force remains constant, how long does it take to move 0.5 m to the left. Use the following parameter values: m A = m B = 70kg and µ s = 0.20.

24 Chapter 15: Homework Problems Beginning from rest when θ = 10 o, a 30 kg child slides with kinetic friction (µ k = 0.1) down the slide which is in the shape of a 2.5 meter circular arc. Determine the child s speed when they reach the bottom. θ r h

25 Chapter 15: Homework Problems A small mass, m, is placed on the surface of a dish at the radius shown. The coefficient of static friction is 0.75 and you may neglect the angular acceleration. Determine the minimum and maximum angular velocities at which the dish can spin without the mass slipping. L = 0.15m and θ = 35 o. ω L m θ

26 Chapter 15: Homework Problems There is a block on a circular cone that is held in place by a cord. The block s mass is 1 kg. If the block is moving at a constant speed of 0.1 m/s, determine the tension in the cord and the normal force on the block. L = 250mm, h = 500mm, and r = 175mm. L z A B ω h r

27 Chapter 15: Homework Problems A 1000 kg car is traveling over the crest of a parabolic hill described by y = 5(1 2x ). When its x-coordinate is 35 m, it is traveling at 15m/s and is increasing its speed by 2m/s 2. Sketch the path of the car. Determine the radius of curvature at the point in question. Treating the car as a particle, determine the force that the road exerts on the car in the tangential and normal directions.